4.5. STRONG INTERACTION POTENTIALS 225
andφ(r)is as in (4.5.36).
In summary, for a particle withNstrong charges and radiusρ, its strong interaction
potential can be written as
(4.5.39)
Φ=gs(ρ)[
1
r−
A
ρφ(r)e−kr]
,
gs(ρ) =N(
ρw
ρ) 3
gs,whereρwis the radius ofw∗-weakton,Ais a dimensionless constant depending on the particle,
φ(r) = 1 +kr, and 1/kis the radius of strong attraction of particles. Phenomenologically, we
take
(4.5.40)
1
k=
10 −^18 cm forw∗−weaktons,
10 −^16 cm for quarks,
10 −^13 cm for neutreons,
10 −^7 cm for atom/molecule.
More specifically, we give the following layered formulas ofstrong interaction poten-
tials for various level of particles as thew∗-weakton potentialΦ 0 , the quark potentialΦq,
the nucleon/hadron potentialΦnand the atom/molecule potentialΦaare given as follows
(Ma and Wang,2014c):
(4.5.41)
Φ 0 =gs[
1
r−
A 0
ρw( 1 +k 0 r)e−k^0 r]
,
Φq=(
ρw
ρq) 3
gs[
1
r−
Aq
ρq( 1 +k 1 r)e−k^1 r]
,
Φn= 3(
ρw
ρn) 3
gs[
1
r−
An
ρn( 1 +knr)e−knr]
,
Φa=N(
ρw
ρa) 3
gs[
1
r−
Aa
ρa( 1 +kar)e−kar]
.
Here,k 0 ,k 1 ,kn,kaare taken as in (4.5.40):
(4.5.42)
1
k 0= 10 −^18 cm,1
k 1= 10 −^16 cm,1
kn= 10 −^13 cm,1
ka= 10 −^10 ∼ 10 −^7 cm.Remark 4.18.For two particles withN 1 ,N 2 strong charges and radiiρ 1 ,ρ 2 , their strong
potential energy is
(4.5.43) V=gs(ρ 1 )gs(ρ 2 )
[
1
r−
B
ρ( 1 +kr)e−kr]
,
whereBandkdepend on the two types of particles, andρdepends onρ 1 andρ 2 withρ=ρ
ifρ=ρ 1 =ρ 2. The strong force between the two particles is
(4.5.44) F=−∇V, V as in (4.5.43).